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Let $(M,g)$ be a Riemannian (possibly non-compact) manifold. Let $f:M\longrightarrow M$ be a harmonic function, that is $\Delta_g(f)=0$. I know that for Euclidean spaces like $(\Bbb R^n, ds^2)$ with the standard metric harmonicity implies smoothness (in particular real analytic). I am wondering if this remains true for other Riemannian Manifolds.

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Assuming you mean $f: M \to \mathbb R$ instead of $f: M \to M,$ then the rule of thumb is that harmonic functions will be as regular as the metric allows them to be. That is, if the metric is smooth, then harmonic functions will be smooth; and if the metric is analytic (with respect to some analytic structure on $M$) then the harmonic functions will be analytic (with respect to this same structure).

This follows from the standard regularity theory of elliptic partial differential equations, since $\Delta_g$ is just a linear elliptic operator when expressed in local coordinates.

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